Biology · Simulator 033
Rabbits breed, foxes eat rabbits, and foxes starve when rabbits are scarce. Change the four rates and watch the populations cycle, with the predator peak always lagging the prey peak. The phase plane shows the same story as a closed loop.
The Lotka–Volterra model describes prey x and predators y:
dx/dt = α x − β x y
dy/dt = δ x y − γ y
The equilibrium is x* = γ/δ, y* = α/β. Away from it, populations orbit in closed cycles. In the plain model, every starting point gives its own cycle and the cycles never die out. The predator peak follows the prey peak by about a quarter cycle, with period near 2π/√(αγ) for small cycles.
Turn on prey carrying capacity K to add crowding (logistic growth, αx(1 − x/K)). Cycles then spiral in towards a steady balance. The equations are integrated with RK4.
Educational model: continuous, well-mixed populations with constant rates. Real ecosystems have seasons, space, other species and randomness.
A pesticide kills some prey and some predators each year, effectively raising γ and lowering α. Using x* = γ/δ and y* = α/β, which population ends up higher on average?
Raising γ raises x* (more prey on average) and lowering α lowers y* (fewer predators). This is Volterra's principle: indiscriminate culling tends to favour the prey. It explains why some pest outbreaks worsened after broad-spectrum spraying removed their predators.